5 pirates managed to seize 100 valuable precious stones of the same size and value.
The decided to divide their plunder in this manner:
1. Decide their number by balloting (number 1 ?C 5)
2. Firstly, no 1 will assign their plan and the 5 of them will vote. If more than half of them agrees, they will carry out the plan. If not <no 1> will get thrown into the sea to feed sharks
3. If no 1 dies, no 2 will take on his duties and the 4 of them will vote. If more than half of them agrees, they will carry out the plan. If not <no 2> will get thrown into the sea to feed sharks
4. Analogize from the above.
Condition: Every pirate is intelligent, able to make sound judgement of the loss and gain and make a decision accordingly.
Question:
How do no 1 assign he plan , so that he is able to achieve the maximum gain?"
Posted on 2003-08-24 02:54:52 by tomorrow
why no guys ?
Posted on 2003-08-24 07:27:25 by tomorrow
I don't think I can fully understand the problem, what kind of plan do they need to make?

Thomas
Posted on 2003-08-24 07:37:53 by Thomas
It seems like No 1 does not want to be shark food, so he needs the majority to vote for his plan. No 1 knows he will vote for his plan, so he just needs two more to vote with him. What kind of plan will No 1 create to ensure two will vote with him?

tomorrow, is that what you mean?
Posted on 2003-08-24 09:19:12 by bitRAKE
It also depends on how the pirates choose.. You say they are intelligent but do they care if the other pirates die? Would they rather have all the stones?

Thomas
Posted on 2003-08-24 09:38:28 by Thomas
you can't gain, they'll vote against you for greater profit :P

this continues at max until 2 are left
Posted on 2003-08-24 10:03:03 by Hiroshimator
Okay I think I got it, *but* I assumed that the pirates' greatest priority is lots of stones (that is, it will always choose for the option where they get the most stones). Also, I assumed they don't like dying so they'd rather agree to getting nothing than die.

I will describe the 5 situations (one plan of each) backwards, starting at only 5 left.

Situation E: Only pirate 5 is left
Pirate 5 keeps all 100 (duh :)).

Score:

Situation D: Pirates 4 and 5 are left, two need to agree
The only plan 4 can make is: give 5 everything. 5 will only agree to that since he can get 100 anyway if he doesn't agree. So 4 will get nothing but life, and 5 gets everthing.

Score:

Situation C: Pirates 3, 4 and 5 are left, two need to agree
5 will never agree to anything but 100 stones, since he can get 100 in situations D and E.
3 needs 4 to agree. 4 will get nothing in situation D, so if he gets nothing in this situation too he can choose to agree or not to, it doesn't make a difference for him then but obviously it does for pirate 3. To ensure pirate 4 agrees, 3 has to give him an advantage over situation D. The minimum advantage is giving him one stone, keeping 99 himself.

Score:

Situation B: Pirates 2, 3, 4 and 5 are left, three need to agree
5 will never agree (can get 100).
4 will agree if he gets more than one stone.
3 will agree only to more than 99 stones, since situation C can give him 99 already.
For 2 to keep living, he needs 3 and 4 to agree (5 will never agree). This is impossible. 3 will only agree when 2 gives him 100 stones, but then 4 won't agree because he gets nothing. But if you give 4 more than one stone, you have only 98 at most left, not enough to make 3 agree. So in any case, 2 will be shark food.

Score:
None

Situation A: All pirates are there, three need to agree
5 will never agree (can get 100).
4 will agree if he gets more than one stone.
3 will agree only to more than 99 stones.
2 will agree to anything, since it's agree or die for him.
1 needs two other people to agree. The best two are 2 and 4. Pirate 2 will always agree, even if 1 gives him nothing. So he doesn't. For 4 to agree, he needs at least two stones, so 1 gives him two stones. Now pirate 1 has 98 stones left.

So conclusion: give pirate 4 two stones, keep 98 yourself and pirates 2 and 4 will agree with pirate 1.

Thomas
Posted on 2003-08-24 10:17:47 by Thomas
Thomas, 5 will never get 100 because Situation C prevents it. :)
Posted on 2003-08-24 11:36:51 by bitRAKE

Thomas, 5 will never get 100 because Situation C prevents it. :)

Isn't that the same as saying 3 never gets 99 because situation A prevents that? Or have I missed something? I worked backwards so none of the situations except for the last will ever happen, if they would happen they would happen in reverse order (A, B, C, D then E).

Thomas
Posted on 2003-08-24 12:06:54 by Thomas
Thomas, you are correct - I was just thrown off by the scoring.
C Score:
Posted on 2003-08-24 12:12:36 by bitRAKE
Ah yes, you're right, I'll fix it in my post..

Thomas
Posted on 2003-08-24 12:15:08 by Thomas
Here is my take on the problem:

Situation B: Pirates 2, 3, 4 and 5 are left, three need to agree
5 willl agree for more than zero (if two dies, he gets nothing in C)
4 will agree if he gets more than one stone.
3 will agree only to more than 99 stones, since situation C can give him 99 already.
For 2 to keep living, he needs 5 and 4 to agree. 5 will agree because if he doesn't then 2 dies and he gets nothing in Situation C. 4 will agree because he get more than he would in C.

Score:
1: dead, 2: 97, 3: 0, 4: 2, 5: 1

Situation A: All pirates are there, three need to agree
5 will agree if he gets more than one
4 will agree if he gets more than two stones
3 will agree if he gets more than zero stones
2 will not agree to anything less than 98
1 needs two other people to agree. The best two are 2 and 4. Pirate 2 will always agree, even if 1 gives him nothing. So he doesn't. For 4 to agree, he needs at least two stones, so 1 gives him two stones. Now pirate 1 has 98 stones left.

Score:
1: 97, 2: 0, 3: 1, 4: 0, 5: 2

It would be cool to program some AI to play this game! :)
Posted on 2003-08-24 20:02:45 by bitRAKE